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Concerning ultrafilters on ultrapowers

Published online by Cambridge University Press:  12 March 2014

J. M. Henle
J. M. Henle*
Affiliation:
Department of Mathematics, Smith College, Northampton, Massachusetts 01063
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Extract

This paper concerns ultrafilters on a cardinal γ extending the filter of λ-closed, unbounded sets, λ < γ. The history of these ultrafilters is closely connected with that of the axiom of determinacy (AD). Solovay noticed first that, under AD, there was such an ultrafilter for γ = ℵ1; λ = ω. Later, Kleinberg found that the existence of such ultrafilters followed from the partition relation γ → (γ)λ+λ . Specific instances of this and more powerful relations on cardinals were then proved from AD by Martin, Kunen, Paris, and others. The axiom of determinacy was recently shown consistent with ZF relative to something less than a supercompact cardinal by Martin and Steel. Solovay's and Kleinberg's results were actually stronger, and we discuss this at the end of the paper. Good references for these results include [K2] and [KM].

We are interested here in the case where γ is the ultrapower of a strong partition cardinal κ (a cardinal satisfying for all α < κ). Such cardinals exist in great abundance assuming AD, and in fact, if sufficiently many cardinals are strong, then AD holds in L[R] [KKMW].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 149 - 151
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[H] Henle, J. M., Researches into the world of κ → (κ) κ , Annals of Mathematical Logic, vol. 17 (1979), pp. 151–169.CrossRefGoogle Scholar
[K1] Kleinberg, E. M., The ℵn are Jónsson cardinals and ℵ ω is a Rowbottom cardinal, Annals of Mathematical Logic, vol. 12 (1977), pp. 229–248.CrossRefGoogle Scholar
[K2] Kleinberg, E. M., Infinitary combinatorics and the axiom of determinateness, Lecture Notes in Mathematics, vol. 612, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
[KM] Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher set theory (Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 669, Springer-Verlag, Berlin, 1978, pp. 99–275.Google Scholar
[KKMW] Kechris, A. S., Kleinberg, E. M., Moschovakis, Y. N., and Woodin, H., The axiom of determinacy, strong partition properties and nonsingular measures, Cabal seminar 77–79, Lecture Notes in Mathematics, vol. 839, Springer-Verlag, Berlin, 1981, pp. 75–99.CrossRefGoogle Scholar
[W] Watro, R. J., Effects of infinite exponent partition properties on Mahlo cardinals, Ph. D. Thesis, State University of New York at Buffalo, Buffalo, New York, 1982.Google Scholar